Some Theoretical Considerations on the Microwave Three-wave Mixing Experiments
Takayoshi Amano
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
Patterson and Doyle demonstrated that optical isomers could be discriminated by using microwave threewave mixing experiments [1]. Apparently their work was
carried out, being stimulated by a theoretical work on
microwave triple resonance by Hirota [2]. Subsequently
their work was followed by several investigations performed by using similar techniques [3–8]. Grabow presented theoretical background on this microwave threewave mixing experiments [9]. Also Lobsiger et al. gave
similar theoretical explanations [8].
For C1 enantiomers, all the dipole moment components along the principal moment of inertia axes are nonzero. The sign of the product of the three components
should be different for the different enantiomers. This
is the reason why the experiments are designed to observe the phenomena that depend on it; triple resonance
or microwave three-wave mixing. In this investigation,
we present some theoretical background of those experiments which is similar to that given by Grabow [9]. However, in the end, we will see a different factor which might
evoke a different perspective on the experiments.
Consider a three level system depicted in Fig. 1.
Assume, for simplicity, that a system is similar to that
used in Patterson and Doyle; the transition between the
levels 1 and 2 is a c-type transition and between the levels 2 and 3 an a-type, and between the levels 1 and 3
a b-type. The time evolution of the wave functions is
generally described as a linear combination of the three
eigenstates with time dependent coefficients such as
X
Ψ(t) =
ai (t)φ(i).
(1)
i
The density matrix ρ is defined to be ρij = ai a∗j as its ma-
trix element. The time dependent density matrix equation is given by
dρ
= [H, ρ],
(2)
dt
where the Hamiltonian H consists of the zero-th order
molecular Hamiltonian and the interaction with the applied external radiation field,
ih̄
H = H0 + H 0 .
(3)
We assume two resonant microwave radiations are applied to excite the transitions between the levels 1 and 2
and the levels 2 and 3. Therefore
H0 = −µE(t)
E(t) = E0 cos ωt + E00 cos ω 0 t
(4)
(5)
where ω = (E2 − E1 )/h̄ and ω 0 = (E3 − E2 )/h̄. As the
transition between the levels 1 and 2 is a c-type transition, the Rabi frequency Ω12 (=µ12 E0 /h̄) can be written as iΩ, and accordingly Ω21 = −iΩ with Ω being a
real quantity by adapting the phase convention used in
microwave spectroscopy ( see, for example, Townes and
Schawlow ). For the transition between the levels 2 and
3, Ω23 = Ω32 = Ω0 , and Ω0 is real.
By using the rotating wave approximation,
0
0
ρ12 = ρ̃12 eiωt , ρ23 = ρ̃23 eiω t , ρ13 = ρ̃13 eiωt eiω t ,
and by defining three Bloch vector components for each
of the three transitions,
uc = ρ̃12 + ρ̃21 , v c = i(ρ̃21 − ρ̃12 ), wc = ρ11 − ρ22
ua = ρ̃23 + ρ̃32 , v a = i(ρ̃32 − ρ̃23 ), wa = ρ22 − ρ33
ub = ρ̃13 + ρ̃31 , v b = i(ρ̃31 − ρ̃13 ), wb = ρ11 − ρ33
the equations of motion can be written as
Ω0
Ω0
Ω0 b c
v , v˙ = − ub , w˙c = −Ωuc − v a
2
2
2
Ω b
Ω c
Ω b a
0
a
0
a
a
u , v˙ = −Ω w + v , w˙ = u + Ω v a
u˙ =
2
2
2
Ω0 c Ω a ˙b
Ω0 c Ω a
Ω
Ω0
˙
b
u =
v − u , v = − u − v , ẇb = − uc + v a .
2
2
2
2
2
2
Suppose first we apply a pulse radiation field which is
resonant to the c-type transition of the time duration of
∆t to excite the transition between the levels 1 and 2.
Then we obtain,
u˙c = Ωwc +
a-type
c-type
b-type
FIG. 1: First shine a pulse radiation which is resonant to
the transition between the levels 1 and 2. Immediately after
this pulse, we apply the second pulse to excite the transition between the levels 2 and 3. Following these preparation
processes, observe free induction decay from the level 3 to 1.
uc (∆t) = wc (0)sin(Ω∆t)
v c (∆t) = wc (0)cos(Ω∆t)
1
wa (∆t) = wc (0)[1 − cos(Ω∆t)] + wa (0)
2
1
wb (∆t) = − wc (0)[1 − cos(Ω∆t)] + wb (0),
2
with all other components to be zero. Following this
pulse, we apply a resonant radiation to excite the a-type
transition between the levels 2 and 3.
follows;
χ0 E0 = N µ̄(ρ̃31 + ρ̃13 ) = N µ̄ub
χ00 E0 = iN µ̄(ρ̃31 − ρ̃13 ) = N µ̄v b
a
u (t) = 0
1
v a (t) = −( wc (0)[1 − cos(Ω∆t)] + wa (0))sinΩ0 (t − ∆t)
2
1 c
a
w (t) = ( w (0)[1 − cos(Ω∆t)] + wa (0))cosΩ0 (t − ∆t)
2
ub (t) = 0
Ω0
v b (t) = wc (0)sin(Ω∆t)sin (t − ∆t)
2
1 a
b
w (t) = − (w (0))(1 − cosΩ0 (t − ∆t)) + wb (0)
2
Ω0
c
c
u (t) = w (∆t)sin(Ω∆t)cos (t − ∆t)
2
v c (t) = 0
1
wc (t) = wc (0)cos(Ω∆t)(1 − cosΩ0 (t − ∆t))
2
+ wc (0)cos(Ω∆t).
Now we see that the polarization of the b-type transition is induced through this “double resonance” process.
The induced polarization is expressed for this transition
as
P = N Tr(µρ)
(6)
It can be written explicitly as
P = N (µ13 ρ31 + µ31 ρ13 )
= N µ̄[(ρ̃31 + ρ̃13 )cosω 00 t − i(ρ̃31 − ρ̃13 )sinω 00 t] (7)
where ω 00 = ω + ω 0 . Here we assume the matrix elements
of the dipole moment, µ13 and µ31 , are real and equal,
and designated as µ̄, but the sign can be either positive or
negative. This induced polarization generates the electric
field and the relationship between the two is given by
P=
00
1
χE0 eiω t + c.c.
2
(8)
where χ is complex susceptibility, χ = χ0 − iχ00 . The
susceptibility is related to the density matrix elements as
[1] D. Patterson and J. M. Doyle.
Phys. Rev. Lett.,
111,023008(2013).
[2] E. Hirota. Proc. Jpn. Acad. B, 88,120-128(2012).
[3] D. Patterson, M. Schnell, and J. M. Doyle. Nature,
497,475-478(2013).
[4] D. Patterson and M. Schnell. Phys. Chem. Chem. Phys.,
16,11114-11123(2014).
[5] V. A. Shubert, D. Schmitz, D. Patterson, J. M. Doyle, and
M. Schnell. Angew. Chem. Int. Ed., 53,1152-1155(2014).
[6] V. A. Shubert, D. Schmitz, and M. Schnell. J. Mol. Spec-
(9)
(10)
Power emitted (or absorbed) is given by
W =E
∂P
ω 00 2 00
=
E χ
∂t
2 0
Therefore we obtain
ω 00
Ω0
E0 µ̄wc (0)sin(Ω∆t)sin (t − ∆t)
2
2
ω 00
Ω0
= (N1 (0) − N2 (0)) E0 µ̄sin(Ω∆t)sin (t − ∆t)
2
2
(11)
The emitted power is maximized by applying sequentially π/2- and π-pulses. As formulated above, the induced polarization from the two-step pumping process
appears to depend on the product of the matrix elements
of the three dipole moment components, µa , µb , and µc .
In the treatment presented above, the Rabi frequencies
are implicitly assumed to be the signed quantities. This
assumption should be examined carefully. In all cases for
two level systems, the sign of the dipole moment does
not matter, and we automatically use the absolute value
of the dipole moment in calculating the Rabi frequency.
In general, the oscillation
frequency of the polarization
p
is given by Ω̄ = ± (ω − ω0 )2 + Ω2 , as the radiation frequency may not be in exact resonance. The sign of Ω̄
can be either positive or negative, but the sign has nothing to do with the sign of the dipole moment. It should
be noted that, in the microwave three-wave mixing experiments done so far, the process was the three-step
sequential two-level processes. If the sign of the product of the three dipole moment components really plays
a role, the power given in eq.(11) indicates emission for
one enantiomer, on the other hand the signal corresponds
to absorption for the other enantiomer. This is not reasonable.
W = N
trosc., 300,31-36(2014).
[7] V. A. Shubert, D. Schmitz, C. Medcraft, A. Krin, D. Patterson, J. M. Doyle, and M. Schnell. J. Chem. Phys.,
142,214201(2015).
[8] S. Lobsiger, C. Perez, L. Evangelisti, K. K. Lehmann, and
B. H. Pate. J. Phys. Chem. Lett., 6,196-200(2015).
[9] J. Grabow.
Angew. Chem. Int. Ed., 52,1169811700(2013).